The correct answer is option 3, 11.

When the coin rolls on the periphery of the disc, its center traces out a path called a cycloid. In one complete revolution, the coin`s center moves along the cycloid once.

To determine the number of times the coin rotates about its center, we need to look at the rotational motion. Since the coin is rolling without slipping, the point on the coin`s circumference that is in contact with the disc is always at rest relative to the disc. This means that the coin`s circumference rotates at the same rate as the disc.

The radius of the coin is 1 cm, and the radius of the disc is 10 cm. Therefore, the circumference of the coin is 2π(1) cm and the circumference of the disc is 2π(10) cm.

To find the number of times the coin rotates about its center, we can divide the circumference of the disc by the circumference of the coin.

Circumference of the disc = 2π(10) = 20π cm

Circumference of the coin = 2π(1) = 2π cm

Number of times the coin rotates about its center = (Circum